Not every flower will produce seeds and in some kinds of plants double flowers produce few seeds or none at all (your hibiscus, or Rose of Sharon, looks like it might be double). The dead-headed flowers would likely not be ripe enough to have mature seeds in any case. Late this fall you should check for dried seed heads and see if any have formed seeds, that is if you haven't deadheaded them all :)
Hibiscus seeds are fairly big, dark and flattish on two sides. They are easy to see. Growing Hibiscus from seed is quite easy but the resulting plants will not be quite the same as the parent. The chances are they may not be as good and it does take at least a couple of years for them to bloom.
Rose of Sharon is relatively easy from cuttings. Cuttings would bloom sooner and would be the only way to exactly duplicate your beautiful white one. However growing things from seed is fun and very satisfying-the resulting plants would be unique and completely your own.

Something else to keep in mind... Rose of Sharon are slow to get going in the beginning. They're kind of lazy, actually. Once they get going though, watch out! As for cuttings blooming sooner, the plants need to be a certain size and age before they'll bloom, going with a cutting isn't always the short - cut it would seem to be. However, cuttings are the way to go to get more of those lovely white floweres (as someone already mentioned). It is quite a beautiful flower.

Seeds should be collected in late fall before the pods (just like any "pod") open up and dispell the seeds. Just as corn in a field gets that tan color and starts to have the tips blacken, same with seed pods. The pods will stay on the plant, but open up during winter until the following year and are very noticable.

I've collected seed from a Bluebird clone for 4 or 5 years or something like that and the only flowers I've ever gotten have all been pink with a red eye.

Googling which cultivars are sterile or fertile will let you know if you have a plant that produces (viable) seed. Sunset Western Garden Book says (1997):
"newer slections are sterile triploids, which have a long blooming season and set few or no seed pods. They include 'Aphrodite', rose pink with deep red eye; 'Diana', pure white; 'Helene', white with deep red eye; and 'Minerva', ruffled lavender pink with reddish purple eye."